
An important feature of the relations above is that they are independent of \(q\) : the exponents vary from model to model, but not the formulae. Relations R1–6 were proved for Bernoulli percolation (i.e. cluster-weight \(q=1\) ) in a milestone paper by Harry Kesten [1]} without any of them being computed, or indeed even be shown to exist (see also [2]}, [3]}, [4]}). For the random-cluster with \(q=2\) , critical exponents were calculated independently [5]}, [6]}, [7]} and were observed to satisfy R1–6. Let us mention that similar relations should hold in all dimensions that are below the so-called upper-critical dimension (with certain values of 2 replaced by the dimension \(d\) ). We refer to a paper by Borgs, Chayes, Kesten and Spencer [8]} (see also [9]}) for a discussion of this phenomenon for Bernoulli percolation.
